Emma Castelnuovo's Intuitive Geometry: A Modern Approach to Teaching Euclidean Geometry
Emma Castelnuovo (1913-2014) was a renowned Italian mathematics educator who published her book \"Intuitive Geometry\" in 1949. In this book, she presented a novel way of teaching and learning Euclidean geometry based on material models, experiments and discoveries. Her method was inspired by her father Guido Castelnuovo and her uncle Federigo Enriques, both eminent mathematicians and reformers of mathematics education in Italy.
Intuitive Geometry was a groundbreaking work that challenged the traditional approach of geometry teaching based on axioms, definitions and proofs. Castelnuovo aimed to develop students' intuition, creativity and reasoning skills by engaging them in active exploration of geometric concepts and properties. She used concrete objects such as paper, cardboard, string, wire and clay to construct models of geometric figures and to perform experiments with them. She also encouraged students to discover geometric facts by themselves through observation, measurement, comparison and generalization.
Castelnuovo's book was widely adopted in Italian schools and influenced many other mathematics educators around the world. However, some critics argued that her method was too informal and lacked rigor and logical structure. They also questioned the relevance of intuitive geometry in the era of digital devices and software that can easily create and manipulate geometric figures on the screen.
In this article, we argue that intuitive geometry by Castelnuovo is still contemporary and valuable for geometry teaching and learning in the digital age. We show that dynamic geometry software (DGS) such as GeoGebra can enhance Castelnuovo's method by allowing students to perform more complex and varied explorations of geometric phenomena. We also show that DGS can help students to bridge the gap between intuition and formalism by providing tools for testing conjectures, finding counterexamples, constructing proofs and generalizing results.
We illustrate our argument with an example of a classroom activity in grade 6 based on Castelnuovo's book. The activity involves constructing a regular pentagon using paper folding and then exploring its properties using GeoGebra. We discuss how this activity fosters students' geometric thinking and understanding by combining material models, dynamic figures and deductive reasoning.
We conclude that intuitive geometry by Castelnuovo is not outdated or obsolete, but rather a modern and innovative approach to teaching Euclidean geometry that can be enriched by the use of digital devices. We suggest that teachers should integrate material models and DGS in their geometry lessons to offer students a rich and meaningful learning experience.
Intuitive geometry by Castelnuovo is based on the idea that geometry is not a set of abstract and arbitrary rules, but a natural and intuitive way of understanding the world around us. Castelnuovo believed that geometry should be taught as a living and evolving science, not as a fixed and dead system. She also believed that geometry should be connected to other disciplines such as art, history, physics and biology, and that it should reflect the cultural and historical diversity of human civilizations.
Castelnuovo's book covers many topics of Euclidean geometry, such as angles, triangles, quadrilaterals, polygons, circles, area, volume, similarity, congruence and symmetry. For each topic, she provides a series of activities that guide students from concrete to abstract levels of thinking. She starts with simple and familiar situations that involve everyday objects and experiences. She then introduces geometric concepts and terms in an informal and intuitive way. She gradually leads students to discover geometric properties and relationships by asking questions, making conjectures and verifying them with experiments. She finally invites students to formalize their findings with definitions, theorems and proofs.
Castelnuovo's book is not a rigid and linear sequence of topics, but rather a flexible and open-ended framework that allows teachers and students to choose their own paths and pace. Castelnuovo encourages teachers to adapt her method to their own contexts and needs, and to create new activities and problems based on their own interests and creativity. She also encourages students to work collaboratively, to communicate their ideas verbally and visually, and to reflect on their own learning processes. 061ffe29dd